Powers of i Calculator
This powers of i calculator works out iⁿ for any whole power — positive, negative, or hundreds of digits long — and shows the remainder that decided it.
The whole subject is four values that go round in a circle: 1, i, −1, −i. Which one you land on depends on a single number, and that number is easier to find than most people are taught.
Why There Are Only Four Answers
Start at 1 and keep multiplying by i — the imaginary unit, called iota in many textbooks:
i⁰ = 1 i¹ = i i² = −1 i³ = −i i⁴ = 1
The fourth step lands back on 1, and from there the same four values repeat forever. The reason is short: i² = −1, so i⁴ = (i²)² = (−1)² = 1. Once you are back at 1 the sequence has no choice but to start again.
Pictured on the complex plane it is a quarter turn each time. Four quarter turns is a full circle, which is the same fact drawn instead of written.
The Remainder Is the Whole Method
Because the pattern repeats every four steps, whole turns round the cycle change nothing and can all be thrown away. What is left is the remainder when the power is divided by 4:
| remainder 0 | iⁿ = 1 |
| remainder 1 | iⁿ = i |
| remainder 2 | iⁿ = −1 |
| remainder 3 | iⁿ = −i |
So i⁴²: 42 divided by 4 leaves 2, giving −1. And i¹²³: 123 leaves 3, giving −i. Nothing else about the power matters at all — not its size, not how it was written, only that one digit.
Negative Powers, and the Trap Inside Them
The same rule applies, with one condition that catches almost everybody: the remainder has to be taken the positive way.
Ask an ordinary division for the remainder of −5 by 4 and you may well get −1. But there is no case for −1 — the four cases are 0, 1, 2 and 3. Adding 4 brings it into range: −1 becomes 3, so i⁻⁵ = −i.
Take the negative remainder as it comes and you get the wrong sign, quietly, with no sign that anything went astray. It is worth checking one case by hand to see that the rule is right: i⁻¹ = 1/i, and multiplying top and bottom by i gives i/i² = i/(−1) = −i — which is exactly what remainder 3 predicts.
The same works all the way down: i⁻² = −1, i⁻³ = i, i⁻⁴ = 1, and then it repeats backwards just as it does forwards.
The Shortcut That Makes Length Irrelevant
Here is the part worth taking away, and it is why a power of any length is no trouble at all.
You never need to divide the whole exponent by 4. A hundred is a multiple of four — and so are a thousand, a million, and every higher place value. That means every digit above the last two sits in a block that divides away exactly, leaving nothing behind.
Only the last two digits of the power change anything.
So for i²⁰²⁹⁶ you can ignore 202 entirely and look at 96: divisible by 4, remainder 0, answer 1. For a power ending in 90, the remainder is 2 and the answer is −1, whether that power has four digits or four hundred.
This calculator will take an exponent hundreds of digits long and answer it instantly for exactly that reason, and it shows you which two digits did the work. Most tools built on ordinary computer numbers cannot go past about sixteen digits before they lose track of the exponent itself — here it is held whole, however long it is.
When There Is a Number in Front
Questions rarely stop at plain i. (2i)⁶ is just as common, and it splits neatly, because a product raised to a power is the product of the powers:
(bi)ⁿ = bⁿ × iⁿ
So (2i)⁶ = 2⁶ × i⁶ = 64 × (−1) = −64. Deal with the number in front and the i separately, then multiply.
A few more: (2i)³ = 8 × (−i) = −8i. (3i)⁴ = 81 × 1 = 81. And with a fraction, (½i)⁴ = 1/16. If what you actually need is to add, subtract, multiply or divide two full numbers of the form a + bi, that is a different job and the complex number calculator does it.
Notice the one real difference this makes. Plain i to a power never grows — the answer is always one of four small values. Put a number in front and it does grow, quickly, which is the only reason this page limits how far it will go when there is a coefficient. Leave that box empty and there is no limit at all.
How to Use This Powers of i Calculator
Two boxes, and usually you only need one.
Put the power in the second box. Any whole number: positive, negative, and as long as you like. Leave the first box empty for plain i, or put a number there for something like (2i)⁶.
| coefficient | The number in front. Empty means plain i |
| to the power | A whole number, of any length, positive or negative |
| ← → ↑ ↓ | Move between the two boxes |
| ± | Flips the sign, for a negative power |
| ⁄ and . | Fraction bar and decimal point, for the coefficient |
| ⌫ / AC | Delete one character, or clear both boxes |
| = | Works it out |
Open the working and you get the division with its remainder, the four-step cycle with your step marked, the two digits that mattered when the power was long, and — for anything small enough — the same answer reached again by multiplying i out one step at a time.
One thing it will not do: a fractional power. That is a genuinely different question with a different kind of answer, since a fractional power of i has several values spread evenly around a circle rather than one. Typing 2.5 gets a refusal here rather than a confident answer to the question you did not ask.
Powers of i FAQ
What are the powers of i?
How do you find i to a large power?
What is i to a negative power?
Do you have to divide the whole exponent by 4?
What is (2i) to the power 6?
Can i be raised to a fractional power?
Four values, one remainder, and only the last two digits doing any work. Everything else about a power of i is decoration.